11

Communication Theory (EC 2252)

Prof.J.B.Bhattacharjee Prof.J.B.Bhattacharjee K.Senthil KumarK.Senthil Kumar

ECE DepartmentECE Department

Rajalakshmi Engineering CollegeRajalakshmi Engineering College

www.Vidyarthiplus.com

Review of Spectral characteristics

Periodic and Non-periodic Signals: A signal is said to be periodic, if it exhibits periodicity. i.e.,

x(t +T)=x(t) , for all values of t. Periodic signal has the property that it is unchanged by a time shift of T. A signal that does not satisfy the above periodicity property is called a non-periodic signal.

Periodic signals can be represented using the Fourier Series. Non-periodic signals can be represented using the Fourier Transform.

Both Fourier series and Fourier Transform deal with the representation of the signals as a combination of sine and cosine waves.

Fourier Series Fourier series: a complicated waveform analyzed

into a number of harmonically related sine and cosine functions

A continuous periodic signal x(t) with a period T may be represented by: x(t)=Σ∞

k=1 (Ak cos kω t + Bk sin kω t)+ A0

Dirichlet conditions must be placed on x(t) for the series to be valid: the integral of the magnitude of x(t) over a complete period must be finite, and the signal can only have a finite number of discontinuities in any finite interval

Fourier Series EquationsThe Fourier series represents a periodic signal Tp in terms of frequency components:

We get the Fourier series coefficients as follows:

The complex exponential Fourier coefficients are a sequence of complex numbers representing the frequency component ω0k.

pT/2ω where,eXx(t) 0k

tikωk

0

p

0

T

tikω

pk dtx(t)e

T

1X

Periodic signals represented by Fourier Series have Discrete spectra.

The Fourier Transform Fourier transform is used for the non-

periodic signals. A Fourier transform converts the signal from the time domain to the spectral domain.

Continuous Fourier Transform:

dfefHth

dtethfH

ift

ift

2

2

Non-periodic signals represented by Fourier transform have Continuous spectra.

Fourier Transform PairsNote: Π stands for rectangular function. Λ stands for triangular function.

99

Introduction to Communication Introduction to Communication SystemsSystems

Communication – Basic process of exchanging information from one location (source) to destination (receiving end).

Refers – process of sending, receiving and processing of information/signal/input from one point to another point.

Source DestinationFlow of information

Figure 1 : A simple communication system

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Electronic Communication System – defined as the whole mechanism of sending and receiving as well as processing of information electronically from source to destination.

Example – Radiotelephony, broadcasting, point-to-point, mobile communications, computer communications, radar and satellite systems.

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ObjectivesObjectives

Communication System – to produce an Communication System – to produce an accurate replica of the transmitted accurate replica of the transmitted information that is to transfer information information that is to transfer information between two or more points (destinations) between two or more points (destinations) through a communication channel, with through a communication channel, with minimum error.minimum error.

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NEED FOR COMMUNICATIONNEED FOR COMMUNICATION Interaction purposes – enables people to Interaction purposes – enables people to

interact in a timely fashion on a global level in interact in a timely fashion on a global level in social, political, economic and scientific areas, social, political, economic and scientific areas, through telephones, electronic-mail and video through telephones, electronic-mail and video conference.conference.

Transfer Information – Tx in the form of audio, Transfer Information – Tx in the form of audio, video, texts, computer data and picture through video, texts, computer data and picture through facsimile, telegraph or telex and internet.facsimile, telegraph or telex and internet.

Broadcasting – Broadcast information to Broadcasting – Broadcast information to masses, through radio, television or teletext.masses, through radio, television or teletext.

1313

Terms Related To CommunicationsTerms Related To Communications Message – physical manifestation produced by the

information source and then converted to electrical signal before transmission by the transducer in the transmitter.

Transducer – Device that converts one form of energy into another form.

Input Transducer – placed at the transmitter which convert an input message into an electrical signal.

Example – Microphone which converts sound energy to electrical energy.

MessageInput

TransducerElectrical

Signal

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Output Transducer – placed at the receiver Output Transducer – placed at the receiver which converts the electrical signal into the which converts the electrical signal into the original message.original message.

Example – Loudspeaker which converts Example – Loudspeaker which converts electrical energy into sound energy.electrical energy into sound energy.

Signal – electrical voltage or current which Signal – electrical voltage or current which varies with time and is used to carry message or varies with time and is used to carry message or information from one point to another.information from one point to another.

ElectricalSignal

OutputTransducer

Message

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Elements of a Communication Elements of a Communication SystemSystem

The basic elements are : Source, The basic elements are : Source, Transmitter, Channel, Receiver and Transmitter, Channel, Receiver and Destination.Destination.

Information Source

TransmitterChannel

Transmission Medium

Receiver Destination

Noise

Figure : Basic Block Diagram of a Communication System

EEE ExclusiveEEE Exclusive

1616

Function of each Element.Function of each Element. Information SourceInformation Source – the communication system – the communication system

exists to send messages. Messages come from exists to send messages. Messages come from voice, data, video and other types of information.voice, data, video and other types of information.

TransmitterTransmitter – Transmit the input message into – Transmit the input message into electrical signals such as voltage or current into electrical signals such as voltage or current into electromagnetic waves such as radio waves, electromagnetic waves such as radio waves, microwaves that is suitable for transmission and microwaves that is suitable for transmission and compatible with the channel. Besides, the compatible with the channel. Besides, the transmitter also do the modulation and encoding transmitter also do the modulation and encoding (for digital signal).(for digital signal).

1717

Block Diagram of a TransmitterBlock Diagram of a Transmitter

5 minutes exercise;5 minutes exercise;Describe the sequence of events that happen at Describe the sequence of events that happen at

the radio waves station during news broadcast?the radio waves station during news broadcast?

ModulatingSignal

AudioAmplifier

ModulatorRF

Amplifier

CarrierSignal

TransmittingAntenna

1818

Channel/MediumChannel/Medium – is the link or path over – is the link or path over which information flows from the source to which information flows from the source to destination. Many links combined will destination. Many links combined will establish a communication networks.establish a communication networks.

There are 5 criteria of a transmission There are 5 criteria of a transmission system; Capacity, Performance, Distance, system; Capacity, Performance, Distance, Security and Cost which includes the Security and Cost which includes the installation, operation and maintenance. installation, operation and maintenance.

2 main categories of channel that 2 main categories of channel that commonly used are; line (guided media) commonly used are; line (guided media) and free space (unguided media)and free space (unguided media)

1919

Receiver Receiver – Receives the electrical signals or – Receives the electrical signals or electromagnetic waves that are sent by the electromagnetic waves that are sent by the transmitter through the channel. It is also transmitter through the channel. It is also separate the information from the received separate the information from the received signal and sent the information to the signal and sent the information to the destination.destination.

Basically, a receiver consists of several stages Basically, a receiver consists of several stages of amplification, frequency conversion and of amplification, frequency conversion and filtering.filtering.

2020

Block Diagram of a ReceiverBlock Diagram of a Receiver

DestinationDestination – is where the user receives the – is where the user receives the information, such as loud speaker, visual information, such as loud speaker, visual display, computer monitor, plotter and printer.display, computer monitor, plotter and printer.

RFAmplifier

Mixer

LocalOscillator

IntermediateFrequencyAmplifier

DemodulatorAudio

AmplifierDestination

Receiving Antenna

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Analog Modulation Analog Modulation

Baseband TransmissionBaseband Transmission Baseband signal is the information either in a Baseband signal is the information either in a

digital or analogue form. digital or analogue form. Transmission of original information whether Transmission of original information whether

analogue or digital, directly into transmission analogue or digital, directly into transmission medium is called baseband transmission.medium is called baseband transmission.

Example: intercom (figure below)Example: intercom (figure below)

MicrophoneVoiceAudio

AmplifierAudio

AmplifierSpeaker

Voice

Wire

2222

Baseband signal is not suitable for Baseband signal is not suitable for long distance communication….long distance communication….

Hardware limitationsHardware limitations Requires very long antennaRequires very long antenna Baseband signal is an audio signal of low frequency. Baseband signal is an audio signal of low frequency.

For example voice, range of frequency is 0.3 kHz to For example voice, range of frequency is 0.3 kHz to 3.4 kHz. The length of the antenna required to 3.4 kHz. The length of the antenna required to transmit any signal at least 1/10 of its wavelength (transmit any signal at least 1/10 of its wavelength (λλ). ). Therefore, L = 100km (impossible!)Therefore, L = 100km (impossible!)

Interference with other wavesInterference with other waves Simultaneous transmission of audio signals will cause Simultaneous transmission of audio signals will cause

interference with each other. This is due to audio interference with each other. This is due to audio signals having the same frequency range and signals having the same frequency range and receiver stations cannot distinguish the signals.receiver stations cannot distinguish the signals.

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ModulationModulation Modulation – defined as the process of modifying a Modulation – defined as the process of modifying a

carrier wave (radio wave) systematically by the carrier wave (radio wave) systematically by the modulating signal.modulating signal.

This process makes the signal suitable for transmission This process makes the signal suitable for transmission and compatible with the channel.and compatible with the channel.

Resultant signal – modulated signalResultant signal – modulated signal

2 types of modulation; Analog Modulation and Digital 2 types of modulation; Analog Modulation and Digital Modulation.Modulation.

Analogue Modulation – to transfer an analogue low pass Analogue Modulation – to transfer an analogue low pass signal over an analogue bandpass channel.signal over an analogue bandpass channel.

Digital Modulation – to transfer a digital bit stream the Digital Modulation – to transfer a digital bit stream the carrier is a periodic train and one of the pulse parameter carrier is a periodic train and one of the pulse parameter (amplitude, width or position) changes according to the (amplitude, width or position) changes according to the audio signal.audio signal.

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Purpose of Modulation Process in Purpose of Modulation Process in Communication Systems Communication Systems

To generate modulated signal that is suitable for To generate modulated signal that is suitable for transmission and compatible with the channel.transmission and compatible with the channel.

To allow efficient transmission – increase transmission To allow efficient transmission – increase transmission speed and distance, eg;speed and distance, eg;

1.1. By using high frequency carrier signal, the information By using high frequency carrier signal, the information (voice) can travel and propagate through the air at (voice) can travel and propagate through the air at greater distances and shorter transmission timegreater distances and shorter transmission time

2.2. Also, high frequency signal is less prone to noise and Also, high frequency signal is less prone to noise and interference. Certain types of modulation have the useful interference. Certain types of modulation have the useful property of suppressing both noise and interferenceproperty of suppressing both noise and interference

3.3. For example, FM use limiter to reduce noise and keep For example, FM use limiter to reduce noise and keep the signal’s amplitude constant. PCM systems use the signal’s amplitude constant. PCM systems use repeaters to generate the signal along the transmission repeaters to generate the signal along the transmission path.path.

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Amplitude Modulation (AM)Amplitude Modulation (AM) Objectives:-Objectives:-

Recognize AM signal in the time domain, frequency Recognize AM signal in the time domain, frequency domain and trigonometric equation formdomain and trigonometric equation form

Calculate the percentage of modulation indexCalculate the percentage of modulation index Calculate the upper sidebands, lower sidebands and Calculate the upper sidebands, lower sidebands and

bandwidth of an AM signal by given the carrier and bandwidth of an AM signal by given the carrier and modulating signal frequenciesmodulating signal frequencies

Calculate the power related in AM signalCalculate the power related in AM signal Define the terms of DSBSC, SSB and VSBDefine the terms of DSBSC, SSB and VSB Understand the modulator and demodulator operationsUnderstand the modulator and demodulator operations

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IntroductionIntroduction ModulationModulation The alteration of the amplitude, phase or frequency of an The alteration of the amplitude, phase or frequency of an

oscillator in accordance with another signal.oscillator in accordance with another signal. Input signal is encoded in a format suitable for transmissionInput signal is encoded in a format suitable for transmission A low frequency information signal is encoded over a higher A low frequency information signal is encoded over a higher

frequency signalfrequency signal Carrier SignalCarrier Signal

Sinusoidal wave,Sinusoidal wave, Modulating Signal/Base bandModulating Signal/Base band

Information signal, Information signal, Modulated WaveModulated Wave

Higher frequency signal which is being modulatedHigher frequency signal which is being modulated Modulation SchemesModulation Schemes

To counter the effects of multi path fading and time-delay spreadTo counter the effects of multi path fading and time-delay spread

tfVv ccc 2sin

tfVv mmm 2sin

2727

Carrier Signal,

Vc

Modulating Signal, Vm

Modulation Schemes

Modulated Signal

VAM

VPM

VFM

2828

Amplitude ModulationAmplitude Modulation Time DomainTime Domain

Frequency DomainFrequency Domain

2929

tfVv mmm 2sin

)2(sin2sin

2sin

tftfV

tfVV

cmm

ccAM

Modulator

Information Signal

Carrier Signal

Output

tfVv ccc 2sin

AM ModulatorAM Modulator

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Amplitude ModulationAmplitude Modulation

Vc

- Vc

Vm

- Vm

Vam

- Vam

3131

Modulation IndexModulation Index Modulation Index, mModulation Index, m

Indicates the amount that the carrier signal is Indicates the amount that the carrier signal is modulated.modulated.

It is an expression of the amount of power in the It is an expression of the amount of power in the sidebands.sidebands.

Modulation level ranges = 0-1 whereModulation level ranges = 0-1 where• 0 = no modulation0 = no modulation• 1 = full modulation1 = full modulation• >1 = distortion>1 = distortion

Vc

Vmm

minmax

minmax

VV

VVm

3232

Modulation IndexModulation Index

Vc

Vmm

3333

Modulation IndexModulation Index

Vmin

Vmin (p-p)

Vmax

Vmax (p-p)

minmax

minmax

VV

VVm

3434

Modulation IndexModulation Index

m = 0 m = 0.5

m = 1

3535

fc

BandwidthBandwidth

2

mVc

2

mVc

VC

fmB

fmfcfmfcB

2

)()(

Bandwidth for AM signal,Bandwidth for AM signal,

fc-fm fc+fm

3636

Power DistributionsPower Distributions

Total transmitted power, PTotal transmitted power, PTT

If R= 1,If R= 1,

USBLSBCT P P P P

2

m 1 P P

2

CT

fc-fm fc+fmfc

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Double Side Band Suppressed Carrier (DSBSC)Double Side Band Suppressed Carrier (DSBSC)

It is a technique where it is transmitting both the It is a technique where it is transmitting both the sidebands without the carrier (carrier is being sidebands without the carrier (carrier is being suppressed/cut)suppressed/cut)

Characteristics:Characteristics: Power content lessPower content less Same bandwidthSame bandwidth Disadvantages - receiver is complex and expensive.Disadvantages - receiver is complex and expensive.

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Single Side Band Single Side Band (SSB)(SSB)

Improved DSBSC Improved DSBSC and standard AM, and standard AM, which waste which waste power and power and occupy large occupy large bandwidthbandwidth

SSB is a process SSB is a process of transmitting of transmitting one of the one of the sidebands of the sidebands of the standard AM by standard AM by suppressing the suppressing the carrier and one of carrier and one of the sidebandsthe sidebands

Advantages:Advantages: Saving powerSaving power Reduce BW by 50%Reduce BW by 50% Increase efficiency, Increase efficiency,

increase SNRincrease SNR DisadvantagesDisadvantages

Complex circuits for Complex circuits for frequency stabilityfrequency stability

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Vestigial Side Band (VSB)Vestigial Side Band (VSB) VSB is mainly used in TV broadcasting for VSB is mainly used in TV broadcasting for

their video transmissions.their video transmissions. TV signal consists ofTV signal consists of

Audio signal – transmitted by FMAudio signal – transmitted by FM Video signal – transmitted by VSBVideo signal – transmitted by VSB

A video signal consists a range of frequency A video signal consists a range of frequency and fmax = 4.5 MHz.and fmax = 4.5 MHz.

If it transmitted using conventional AM, the If it transmitted using conventional AM, the required BW is 9 MHz (BW=2fm). But required BW is 9 MHz (BW=2fm). But according to the standard, TV signal is according to the standard, TV signal is limited to 7 MHz onlylimited to 7 MHz only

So, to reduce the BW, a part of the LSB of So, to reduce the BW, a part of the LSB of picture signal is not fully transmitted.picture signal is not fully transmitted.

4040

Vestigial Side Band (VSB)Vestigial Side Band (VSB)

The frequency spectrum for the TV signal / VSB:The frequency spectrum for the TV signal / VSB:

LowerVideoBands

UpperVideoBands

Total TV signal bandwidth = 7 MHz

Video

Carrier

Audio

Carrier

4.5 MHz

UpperAudioBands

LowerAudioBands

1.25 6.755.75 7.06.250

f (MHz)

4141

Modulator CircuitsModulator Circuits

R1

R2

R3

Diode

C L

Modulating Signal

Output

Carrier

A

B

C D

E

4242

Modulator CircuitsModulator Circuits

A. Modulating Signal

B. Carrier

C. Sum of carrier and

modulating signal

D. Diode current

E. AM output across

tuned circuit

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DemodulatorDemodulator

R1

Diode

C1

C’

R’AM

Signal

A B C

4444

DemodulatorDemodulator A. AM signal

B. Current pulses

through diode

C. Demodulating signal

D. Modulating signal

4545

Frequency Modulation (FM)Frequency Modulation (FM) Objectives:-Objectives:-

Recognize FM signal in the time domain, frequency Recognize FM signal in the time domain, frequency domain and trigonometric equation formdomain and trigonometric equation form

Calculate the percentage of modulation indexCalculate the percentage of modulation index Calculate the upper sidebands, lower sidebands and Calculate the upper sidebands, lower sidebands and

bandwidth of an FM signal by Carsons’s Rule and bandwidth of an FM signal by Carsons’s Rule and Bessel Function TableBessel Function Table

Calculate the power related in FM signalCalculate the power related in FM signal Understand the modulator and demodulator of FMUnderstand the modulator and demodulator of FM

4646

IntroductionIntroduction

FM is the process of varying the frequency of a FM is the process of varying the frequency of a carrier wave in proportion to a modulating signal.carrier wave in proportion to a modulating signal.

The amplitude of the carrier is kept constant while its The amplitude of the carrier is kept constant while its frequency is varied by the amplitude of the frequency is varied by the amplitude of the modulating signal. modulating signal.

In all types of modulation, the carrier wave is varied In all types of modulation, the carrier wave is varied by the AMPLITUDE of the modulating signal. by the AMPLITUDE of the modulating signal.

FM signal does not have an envelope, therefore the FM signal does not have an envelope, therefore the FM receiver does not have to respond to amplitude FM receiver does not have to respond to amplitude variations variations it can ignore noise to some extent. it can ignore noise to some extent.

4747

Frequency Modulation

4848

Frequency ModulationFrequency Modulation

The importance features about FM waveforms The importance features about FM waveforms are:are: The frequency variesThe frequency varies The rate of change of carrier frequency changes is The rate of change of carrier frequency changes is

the same as the frequency of the information signalthe same as the frequency of the information signal The amount of carrier frequency changes is The amount of carrier frequency changes is

proportional to the amplitude of the information proportional to the amplitude of the information signalsignal

The amplitude is constantThe amplitude is constant

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Carrier SignalCarrier Signal Sinusoidal waveSinusoidal wave

Modulating Signal/Base bandModulating Signal/Base band Information signalInformation signal

Modulated WaveModulated Wave Higher frequency signal which is being modulatedHigher frequency signal which is being modulated

Where Where

tfVv ccc 2sin

tfVv mmm 2sin

Frequency ModulationFrequency Modulation

)2sin2(cos tftfVv mccFM

mf

KVm

2

5050

Frequency ModulationFrequency Modulation Time DomainTime Domain

Frequency DomainFrequency Domain

5151

FM ModulatorFM Modulator

5252

FM ModulatorFM Modulator

tfVv mmm 2sin

tfVv ccc 2sin

ModulatorInformation Signal

Carrier Signal

Output

)2sin2(cos tftfVv mccFM

5353

FrequencyFrequency Carrier FrequencyCarrier Frequency

As in FM system, carrier frequency in FM systems As in FM system, carrier frequency in FM systems must be higher than the information signal frequency.must be higher than the information signal frequency.

Maximum FrequencyMaximum Frequency

Minimum FrequencyMinimum Frequency

Carrier SwingCarrier Swing

ffcf min

ffcf xma

ffcs 2

5454

Modulation IndexModulation Index Modulation Index, m @ Modulation Index, m @ ββ

Indicates the amount that the carrier signal is Indicates the amount that the carrier signal is modulated.modulated.

It is an expression of the amount of power in the It is an expression of the amount of power in the sidebands.sidebands.

Modulation level ranges = Modulation level ranges = 0 0 –– Where Where

• ΔΔf = fd = frequency deviationf = fd = frequency deviation• fm = modulating frequencyfm = modulating frequency• Vm = amplitude of modulating signalVm = amplitude of modulating signal

fm

fm

2kVm

f

5555

Modulation IndexModulation Indexββ = 1 = 1

ββ = 5 = 5

5656

Modulation IndexModulation Index

ββ = 25 = 25

5757

Modulation IndexModulation Index

5858

BandwidthBandwidth

Using Bessel Function, the bandwidth for Using Bessel Function, the bandwidth for FM signal,FM signal,

n = number of pairs of the significant n = number of pairs of the significant sidebandssidebands

fm = the frequency the modulating signalfm = the frequency the modulating signal

nfmBW 2

5959

BandwidthBandwidth

Using Carson’s Rule, to estimate the Using Carson’s Rule, to estimate the bandwidth for an FM signal transmission.bandwidth for an FM signal transmission.

ΔΔf f = peak frequency deviation= peak frequency deviation

ffm(max)m(max) = highest modulating signal frequency = highest modulating signal frequency

)(2(max)m

ffBW

6060

Power DistributionsPower Distributions FM transmitted power, PFM transmitted power, PFMFM

wherewhere

2R

P

R

V P

2C

2rms

FM

2

V Vrms

Narrowband FM and Wideband FM Narrowband FM has only a single pair of significant

sidebands.  The value of modulation index β <1.

Wideband FM has a large number  (theoretically infinite) number of sidebands. The value of modulation index β >=1.

Generation of Narrowband FM (NBFM)

The modulator splits the carrier into two paths. One path is direct. The other path contains a -90 degree phase shift unit and a product modulator. The difference between the signals in the two paths produces the NBFM signal.

INTEGRATOR

-90 PHASESHIFTER

PRODUCTMODULATOR

Σ_

+

NBFM WAVE

CARRIER WAVEMODULATING

WAVE

)2sin2(cos tftfVv mccFM

)2sin()2sin()2(cos

,1

tftfVtfVv

havewethenIf

mccccNBFM

Frequency Modulators

A frequency modulator is a circuit that varies carrier frequency in accordance with the modulating signal.

There are two types of frequency modulator circuits.

(1) Direct FM: Carrier frequency is directly varied by the message through voltage-controlled oscillator. Eg: Varactor diode modulator.

(2) Indirect FM: Generate NBFM first, then NBFM is frequency multiplied for targeted Δf. Eg: Armstrong modulator

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FM FM Varactor Modulator

The Operation of the Varactor Modulator

The info signal is applied to the base of the input transistor and appears amplified and inverted at the collector.

This low freq signal passes through the RF choke (L1) and is applied across the varactor diode.

Varactor diode behaves as voltage controlled capacitor.

When low reverse biased voltage is applied, more capacitance is generated and thus decrease the frequency.

When high reverse biased voltage is applied, less capacitance is generated and thus increase the frequency.

The varactor diode changes its capacitance in sympathy with the info signal and therefore changes the total value of the capacitance in the tuned circuit.

The changing value of capacitance causes the oscillator freq to increase and decrease under the control of the information signal.

The output is therefore an FM signal.

Armstrong of indrect FM generation

In this method the message signal is first subjected to NBFM modulator using a crystal-controlled oscillator for generating carrier.

Crystal control provides frequency stability.

The NBFM wave is next multiplied in frequency by using a frequency multiplier so as to produce the desired wideband FM.

Frequency Demodulator The FM demodulating circuits used to recover

the original modulating signal.

Any circuit that will convert a frequency variation in the carrier back into a proportional voltage variation can be used to demodulate or detect FM signals.

A popular method used for FM demodulation is the Frequency discriminator.

Frequency discriminator

Output of the Frequency discriminator

The Frequency discriminator circuit consists of the slope ciruit followed by the envelope detector.

The slope circuit converts the instantaneous frequency variations of the FM input signal to instantaneous amplitude variations.

These amplitude variations are rectified by the envelope detector to provide a DC output voltage which varies in amplitude and polarity with the input signal frequency.

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FM vs AM:FM vs AM:

AdvantagesAdvantages DisadvantagesDisadvantages

Better noise Better noise immunityimmunityRejection of Rejection of interfering signals interfering signals because of capture because of capture effecteffectBetter transmitter Better transmitter efficiencyefficiency

Excessive use of Excessive use of spectrumspectrumMore complex and More complex and costly circuitscostly circuits

Review of Probability Sample Space ： the space of all possible outcomes

(δ) Event ： a collection of outcomes ： subset of δ Probability ： a “measure” assigned to the events of a

sample space with the following properties：1. for all event A in S2. 3. If A and B are mutually exclusive,

Theorem:

The Conditional probability of an event A given the occurrence of event B is

0)(AP1)( SP

)()()( BPAPBAP

)()()()( BAPBPAPBAP

)(

)()|(

BP

BAPBAP

Two events A and B are independent if

Random Variables A rule which assigns a numerical value to

each possible outcomes of a chance experiment.

If the experiment is flipping a coin. Then a random variable X can be defined as :

)()()( BPAPBAP

S1 H X(S1)=1

S2 T X(S2)=-1

Cumulative Distribution Function (CDF) ≜ Properties of CDF： 1.

2.

3.

Probability Density Function (PDF) ≜ Properties of PDF ： ,

,

)(xFX }{Prob xX

0)(,1)(,1)(0 XXX FFxF

).()(lim)( 00

i.e. right, from continuous is xFxFxF XXxx

X

. of function ingnondecreasa is )( xxFX

)(xf X dttfxFdx

xdF x

XXX

)()( )(

0)( xf X 1)(

dxxf X

dfxfxFxFxXxPx

x XXX )()()()( 2

11221

Random Processes: A random process is a mapping from the sample space to an ensemble of time functions.

X1(t)

X2(t)

XN(t)

Sample function

t

The totality of all sample functions is called

an ensemble

For a specific timeX(tk) is a random variable

A random process X(t) is a Gaussian process if for all n and for all (t1 t2 ... tn), the sequence of random variables { X(t1), X(t2)... X(tn) } has a jointly Gaussian density function.

Central limit theorem The sum of a large number of independent

and identically distributed(i.i.d) random variables getting closer to Gaussian distribution.

Thermal noise can be closely modeled by Gaussian process.

Gaussian process

Property 1 For Gaussian process, knowledge of the

mean(m) and covariance(C) provides a complete statistical description of process.

Property 2 If a Gaussian process X(t) is passed through

a LTI system, the output of the system is also a Gaussian process. The effect of the system on X(t) is simply reflected by the change in mean(m) and covariance(C) of X(t).

Noise Theory Shot noise: It results from the shot effect in the

amplifying devices and active device. It is caused by random variation in the arrival of electrons (or holes) at the output of the devices.

For diode, the rms shot noise current is given by:

systemofbandwidthδ

currentdiodedirecti

electronofchargee

noiseshotrmsi

δ2eii

f

p

n

fpn

Thermal noise is the electrical noise arising from the random motion of electrons in a conductor. The noise power generated by a resistor is given by:

systemofbandwidthδ

etemperaturabsoluteT

constantsBoltzmann'k

powernoiseP

kTδP

f

n

fn

White noise: It is the idealized form of noise, whose spectrum is independent of the operating frequency. The power spectral density of white noise w(t) is Sw(f)=N0 /2. The autocorrelation Rw(t) of white noise is an impulse as shown below.

Sw(f)

Rw()

)(2

N 0

20N

f

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Narrow band noise (Ideal case) Narrow band noise (Ideal case)

w(t)w(t) n(t)n(t) filtered noise is narrow-band noisefiltered noise is narrow-band noise n(t) = nn(t) = nII(t)cos(2(t)cos(2ffCCt) - nt) - nQQ(t)sin(2(t)sin(2ffCCt)t)

• where nwhere nII(t) is inphase, n(t) is inphase, nQQ(t) is quadrature component (t) is quadrature component filtered signal x(t)filtered signal x(t) x(t) = s(t) + n(t)x(t) = s(t) + n(t) - Average Noise Power = N- Average Noise Power = N00BBTT

BPF

Noise Figure Consider a signal source. The signal to noise

ratio (SNR) available from the source is given by:

Consider that the source is connected to an amplifier with gain G. Since all amplifiers contribute noise, the available output SNR will be less than the SNR of the source.

systemofbandwidthδ

etemperaturabsoluteT

constantsBoltzmann'k

source thefrompowersignalP

/kTδP(S/N)

f

si

fsiin

The noise power at the output of the amplifier will be

The noise factor F is defined as :

When noise factor is expressed in decibels, it is called noise figure.

Noise figure = (F) dB = 10logF

f

no

si

no

f

si

GkT

P

P

P

kT

P F

outputat ratiopower S/N available

inputat ratiopower S/N availableF

G

fno GkT P

The noise power expressed in terms of a temperature is callled Noise Temperature.

If the amplifier noise is Pna , then the equivalent noise temperature Te of the amplifier is given by the equation k/P Te fna

0

0ff0fna

f0na

1)T-(F Te

1)T-(F k/1)kT-(F k/P Te

as written becan re temperatunoise The

1)kT-(F P Since

AM SUPERHETERODYNE RECEIVER

RF section: It generally consists of a pre-selector and an amplifier stage. The pre-selector is a broad tuned band-pass filter with adjustable center frequency that is tuned to the desired carrier frequency. The other functions of the RF section are detecting, band limiting and amplifying the received RF signals.

Mixer/converter section: It is the stage of down-converts the received RF frequencies to intermediate frequencies (IF) which are simply frequencies that fall somewhere between the RF and information frequencies, hence the name intermediate. This section also includes a local oscillator (LO).

IF Section: IF or intermediate frequency section is the stage where its primary functions are amplification and selectivity.

AM detector Section: AM detector section is the stage that demodulates the AM wave and converts it to the original

information signal.

Audio section: Audio section is the stage that amplifies the recovered information.

8888

Performance of CW Modulation Performance of CW Modulation SystemsSystems

Introduction Introduction - Receiver Noise (Channel Noise) : - Receiver Noise (Channel Noise) :

additive, White, and Gaussian additive, White, and Gaussian Receiver Model Receiver Model 1. RX Model 1. RX Model

Sw(f)

Rw()

)(2

N 0

2

N 0

f

N0 = KTe where K = Boltzmann’s constantN0 = KTe where K = Boltzmann’s constant Te = equivalent noise Temp. Te = equivalent noise Temp. Average noise power per unit bandwidth Average noise power per unit bandwidth

SNR The signal x(t) available for demodulation is defined by

The output signal-to-noise ratio (SNR)O is defined as the ratio of the average power of the demodulated message signal to the average power of the noise, both measured at the receiver output.

The channel signal-to-noise ratio, (SNR)C is defined as the ratio of the average power of the modulated signal to the average power of the channel noise in the message bandwidth, both measure at the receiver input.

For the purpose of comparing different CW modulation systems, we normalize the receiver performance by dividing (SNR)O by (SNR)C. This ratio is called figure of merit for the receiver and is defined as

)()()( tntstx

C

O

SNR

SNR

)(

)(meritofFigure

9090

Noise in DSB-SC Receivers

Let’s consider the case of DSB-SC. The expression for the modulated signal is given as

The carrier wave is statistically independent of the message signal. The average power of DSB-SC modulated component of s(t) is

+ BPFx(t) Product

modulator

y(t)DSB-SC

signal s(t)

Noisew(t)

LPFv(t)

LocalOscillator

cos(wct)

Coherent detector

)()2cos()( tmtfAts cC

2

2mc PA

With a noise PSD of N0/2 the average noise power in the message bandwidth W equals WN0 (baseband scenario).

Pm is the power of the message. Hence we have

Finding an expression for (SNR)O, we have

0

2

C 2(SNR)

WN

PA mc

)()()( tntstx tftntftntmtfA cQcIcc 2sin)(2cos)()(2cos

tftntftntmAtntmA

tftxtv

cQcIcIc

c

4sin)(2

14cos)()(

2

1)(

2

1)(

2

2cos)()(

Output of the LPF is

The power of the signal component at the receiver output is . The average power of the filtered noise is 2WN0.

The average noise power at the receiver output is

Hence we have,

)(2

1)(

2

1)( tntmAty Ic

4/2mPAC

elsewhere

WfWffSffSfSfS cNcN

NN QI ,0

),()()()(

00

2

2

12

2

1WNWN

0

2

0

2

22/

4/

WN

PA

WN

PA(SNR) mcmc

O,DSB-SC 1)(

)(meritofFigure

C

O

SNR

SNR

Noise in AM receiver using envelope detection

The expression for AM signal is given as

where it is assumed that

The average power of the carrier in the AM signal s(t) is

The average power of the information bearing component

is

Average power of the full AM signal s(t) is

tftmkAts cac 2cos)(1)(

1)( tmka

+ BPFx(t) Envelope

Detector

y(t)AM signal

s(t)

Noisew(t)

.2/2CA

tftmkA cac 2cos)( 2/22maC PkA

2/)1( 22maC PkA

Hence, the channel signal to noise ratio for AM is

Finding an expression for (SNR)O, we have

0

22

, 2

1)(

WN

PkASNR maC

AMC

)()()( tntmkAty IaC

0

22

, 2)(

WN

PkASNR mC

AMOa

m

m

AMC

O

Pk

Pk

SNR

SNRMeritofFigure

a

a

2

2

1)(

)(

)()()( tntstx

)2sin()()2cos()()()( tftntftntmkAAtx cQcIaCC

)(ofenvelope)( txty

Threshold Effect When carrier-to-noise ratio is small as compared

to unity the noise term dominates the performance of the envelope detector and is completely different. Representing the narrowband noise n(t) in terms of its envelope and phase, we have

The phasor diagram for x(t) = s(t) + n(t) becomes

)(2cos)()( ttftrtn c

Resultant y(t)

r(t)

)(t

)(

1

tmk

A

a

C

)(cos)(1 ttmkA aC

)

(si

n)

(1

tt

mk

Aa

C

The noise envelope is used as a reference here due to its dominance. Here it is assumed that Ac is small as compared to r(t). If we neglect the quadrature component of the signal with respect to the noise we have

Hence, when carrier-to-noise ratio is small the detector has no component that is strictly proportional to the message signal m(t). Recalling that is uniformly distributed over radians. Hence, it follows that we have a complete loss of information at the detector output (as expected value will be zero). This loss of information m(t) at the output of the envelope detector is called the threshold effect.

)(cos)()(cos)()( ttmkAtAtrty aCC

)(t

Pre-emphasis and De-emphasis FM results is an unacceptably low SNR at the high

frequency end of the message spectrum. To offset this undesirable occurrence, pre-emphasis and de-emphasis technique is used.

Pre-emphasis consists in artificially boosting the spectral components in the higher part of the message spectrum. This is accomplished by passing message signal m(t) , through the pre-emphasis filter, denoted Hpe(f) . The pre-emphasized signal is used to frequency modulate the carrier at the transmitting end.

In the receiver, the inverse operation, de-emphasis, is performed. This is accomplished by passing the discriminator output through a filter, called the de-emphasis filter, denoted Hde(f ) .

9898

Pre-emphasis and de-emphasis in FM

P.S.D. of noise at FM Rx output

P.S.D. of typical message signal

Commercial FM radio 에서 사용

otherwise 0

2f

A

fN

(f)S

output tordiscrimina the at (t)n noise of P.S.D

WfW- , )(

1)(

2

C

2

0

Nd

d

T

pe

de

B

fHfH

Information theoryInformation theory

What is What is information theoryinformation theory ? ? Information theoryInformation theory is needed to enable the is needed to enable the

communication system to carry information communication system to carry information (signals) from sender to receiver over a (signals) from sender to receiver over a communication channelcommunication channel• it deals with mathematical modelling and analysis it deals with mathematical modelling and analysis

of a communication systemof a communication system• its major task is to answer to the questions of its major task is to answer to the questions of

signal compressionsignal compression and data and data transfer rate.transfer rate. Those answers can be found and solved by Those answers can be found and solved by

entropyentropy and and channel capacitychannel capacity

Information is a measure of uncertainty. The less is the probability of occurrence of a certain message, the higher is the information.

Since the information is closely associated with the uncertainty of the occurrence of a particular symbol, When the symbol occurs the information associated with its occurrence is defined as:

k'.' symbolby carriedn informatio theis I and

k'' symbol of occurrence ofy probabilit theis P where

)log(P- )P

1(log I

k

k

kk

k

EntropyEntropy EntropyEntropy is defined in terms of probabilistic is defined in terms of probabilistic

behaviour of a source of informationbehaviour of a source of information In information theory the source output In information theory the source output

are discrete random variables that have a are discrete random variables that have a certain fixed finite alphabet with certain certain fixed finite alphabet with certain probabilitiesprobabilities Entropy is an average information content for Entropy is an average information content for

the given source symbol. (bits/message)the given source symbol. (bits/message)

1

02 )

1(log

K

k kk ppH

Rate of information:

If a source generates at a rate of ‘r’ messages per second, the rate of information ‘R’ is defined as the average number of bits of information per second.

‘H’ is the average number of bits of information per message. Hence

R = rH bits/sec

Source CodingSource Coding Source codingSource coding (a.k.a lossless data (a.k.a lossless data

compression) means that we will remove compression) means that we will remove redundant information from the signal prior the redundant information from the signal prior the transmission. transmission.

Basically this is achieved by assigning short Basically this is achieved by assigning short descriptions to the most frequent outcomes of descriptions to the most frequent outcomes of the source output and vice versa.the source output and vice versa.

The common source-coding schemes are The common source-coding schemes are prefix coding, huffman coding, lempel-ziv prefix coding, huffman coding, lempel-ziv coding.coding.

Source Coding TheoremSource Coding Theorem Source coding theoremSource coding theorem states that the output of states that the output of

any information source having entropy H units per any information source having entropy H units per symbol can be encoded into an alphabet having N symbol can be encoded into an alphabet having N symbols in such a way that the source symbols symbols in such a way that the source symbols are represented by code words having a weighted are represented by code words having a weighted average length average length not less than H/logNnot less than H/logN..

Hence source coding theorem says that encoding Hence source coding theorem says that encoding of messages from a source with entropy H can be of messages from a source with entropy H can be done, bounded by the fundamental information done, bounded by the fundamental information theoretic limitation that the theoretic limitation that the Minimum average Minimum average number of symbols/message isnumber of symbols/message is H/logN.H/logN.

Source coding exampleSource coding example

Prefix coding Prefix coding has an important feature has an important feature that it is always uniquely decodable that it is always uniquely decodable and it also satisfies Kraft-McMillan and it also satisfies Kraft-McMillan (see formula 10.22 p. 624) inequality (see formula 10.22 p. 624) inequality term term

Prefix codes can also be referred to as Prefix codes can also be referred to as instantaneous codes, meaning that instantaneous codes, meaning that the decoding process is achieved the decoding process is achieved immediatelyimmediately

Shannon-Fano Coding: In Shannon–Fano coding, the symbols are arranged in order from most probable to least probable, and then divided into two sets whose total probabilities are as close as possible to being equal. All symbols then have the first digits of their codes assigned; symbols in the first set receive "0" and symbols in the second set receive "1".

As long as any sets with more than one member remain, the same process is repeated on those sets, to determine successive digits of their codes. When a set has been reduced to one symbol, of course, this means the symbol's code is complete and will not form the prefix of any other symbol's code.

Huffman Coding: Create a list for the symbols, in decreasing order of probability. The symbols with the lowest probability are assigned a ‘0’ and a ‘1’.

These two symbols are combined into a new symbol with the probability equal to the sum of their individual probabilities. The new symbol is placed in the list as per its probability value.

The procedure is repeated until we are left with 2 symbols only for which 0 and 1 are assigned.

Huffman code is the bit sequence obtained by working backwards and tracking sequence of 0’s and 1’s assigned to that symbol and its successors.

Lempel-Ziv Coding: A drawback of Huffman code is that knowledge of probability model of source is needed. Lempel-Ziv coding is used to overcome this drawback.

while Huffman’s algorithm encodes blocks of fixed size into binary sequences of variable length, Lempel-Ziv encodes blocks of varying length into blocks of fixed size.

Lempel-Ziv coding is performed by parsing the source data into segments that are the shortest subsequences not encountered before.

Mutual InformationMutual Information

Consider a communication system with a source of entropy Consider a communication system with a source of entropy H(X). The entropy on the receiver side be H(Y).H(X). The entropy on the receiver side be H(Y).

H(X|Y) and H(Y|X) are the conditional entropies, and H(X,Y) H(X|Y) and H(Y|X) are the conditional entropies, and H(X,Y) is the joint entropy of X and Y.is the joint entropy of X and Y.

Then the Mutual information between the source X and the Then the Mutual information between the source X and the receiver Y can be expressed as:receiver Y can be expressed as:

I(X,Y) = H(X) - H(X|Y) I(X,Y) = H(X) - H(X|Y)

H(X) is the uncertainty of source X and H(X/Y) is the H(X) is the uncertainty of source X and H(X/Y) is the uncertainty of X given Y. Hence the quantity H(X) - H(X|Y) uncertainty of X given Y. Hence the quantity H(X) - H(X|Y) represents the reduction in uncertainty of X given the represents the reduction in uncertainty of X given the knowledge of Y. Hence I(X,Y) is termed mutual information.knowledge of Y. Hence I(X,Y) is termed mutual information.

Source X

Channel Receiver Y

Channel CapacityChannel Capacity

Capacity in the channel is defined as a Capacity in the channel is defined as a intrinsic ability of a channel to convey intrinsic ability of a channel to convey information.information.

Using mutual information the channel Using mutual information the channel capacity of a discrete memoryless channel is capacity of a discrete memoryless channel is the the maximummaximum average mutual information in average mutual information in any single use of channel over all possible any single use of channel over all possible probability distributions.probability distributions.

Thus Channel capacity C=max( I(X,Y) ).Thus Channel capacity C=max( I(X,Y) ).

Shannon’s Channel Coding theoremShannon’s Channel Coding theorem

The Shannon theorem states that given a noisy channel The Shannon theorem states that given a noisy channel with channel capacity with channel capacity CC and information transmitted at a and information transmitted at a rate rate RR, then if , then if RR < < CC there exist codes that allow the there exist codes that allow the probability of error at the receiver to be made arbitrarily probability of error at the receiver to be made arbitrarily small. This means that theoretically, it is possible to transmit small. This means that theoretically, it is possible to transmit information nearly without error at any rate below a limiting information nearly without error at any rate below a limiting rate, rate, CC..

The converse is also important. If The converse is also important. If RR > > CC, an arbitrarily small , an arbitrarily small probability of error is not achievable. All codes will have a probability of error is not achievable. All codes will have a probability of error greater than a certain positive minimal probability of error greater than a certain positive minimal level, and this level increases as the rate increases. So, level, and this level increases as the rate increases. So, information cannot be guaranteed to be transmitted reliably information cannot be guaranteed to be transmitted reliably across a channel at rates beyond the channel capacity.across a channel at rates beyond the channel capacity.

Shannon-Hartley theorem or Information Capacity Theorem

An application of the channel capacity concept to an additive white Gaussian noise channel with B Hz bandwidth and signal-to-noise ratio S/N is the Information Capacity Theorem.

It states that for a band-limited Gaussian channel operating in the presence of additive Gaussian noise, the channel capacity is given by

C = B log2(1 + S/N) where C is the capacity in bits per second, B is the

bandwidth of the channel in Hertz, and S/N is the signal-to-noise ratio.

Band width and SNR tradeoff As the bandwidth of the channel increases, it

is possible to make faster changes in the information signal, thereby increasing the information rate.

However, as B , the channel capacity does not become infinite since, with an increase in bandwidth, the noise power also increases.

As S/N increases, one can increase the information rate while still preventing errors due to noise.

For no noise, S/N and an infinite information rate is possible irrespective of bandwidth.

Implications of the Information Capacity Implications of the Information Capacity TheoremTheorem

Rate distortion theory Rate distortion theory is the branch of information

theory addressing the problem of determining the minimal amount of entropy or information that should be communicated over a channel such that the source can be reconstructed at the receiver with a given distortion.

Rate distortion theory can be used for the given below situations:

1. Source coding in which the coding alphabet cannot exactly represent the source information.

2. when the information is to be transmitted at a rate greater than channel capacity.

Lower the bit rate R by allowing some acceptable distortion D of the signal

Rate Distortion Function: The functions that relate the rate and

distortion are found as the solution of the following minimization problem.

In the above equation, I(X,Y) is the Mutual information.

Rate distortion function for Gaussian memory-less source

If Px(X) is Gaussian, variance is and if we assume that successive samples of the signal x are stochastically independent, we find the following analytical expression for the rate distortion function.

A Plot of the Rate distortion function for Gaussian source

Lossy Source Coding

Lossy source coding is the representation of the source in digital form with as few bits as possible while maintaining an acceptable loss of information.

In lossy source coding, the source output is encoded at a rate less than the source entropy.

Hence there is reduction in the information content of the source.

Eg: It is not possible to digitally encode an analog signal with a finite number of bits without producing some distortion.